Understand the idea.
Describe the whole distribution
Describe shape by looking across the whole distribution: symmetry, clusters, gaps and unusual values. Positive skew means a longer tail towards high values; negative skew means a longer tail towards low values.
The tail gives skew its name, even when most observations lie at the other end. A high extreme can pull the mean upwards while the median stays near the bulk of the data.
Centre needs spread
The mean shares the total equally across observations. The median is the middle ordered value, averaging the middle pair when the count is even. Range is maximum minus minimum and measures the full observed span.
Report centre and spread together, using the variable's units. Two groups can share a mean while one is tightly concentrated and the other widely scattered. A higher group mean also need not imply every individual value is higher.
See the method in action.
A skewed set
Find mean, median and range for waits 2, 3, 3, 4, 18 minutes.
Worked solution
- 1
Calculate the mean wait.
The total waiting time is shared across five observations.
- 2
Locate the median.
The data are already ordered, so the third of five values is the middle.
- 3
Calculate the range.
The range spans the longest and shortest recorded waits.
- 4
Explain the difference in centre.
The eighteen-minute wait raises the mean to six minutes, above the three-minute median, showing the influence of the high tail.
Mean 6 min; median 3 min; range 16 min. The large wait pulls up the mean.
Equal centre, unequal spread
Compare A: 9, 10, 11 and B: 2, 10, 18.
Worked solution
- 1
Calculate each mean.
The two totals and sample sizes are equal.
- 2
Calculate each range.
The full span differs even though the means agree.
- 3
Compare centre and variability together.
Both groups centre on ten, but B has a much larger spread. The equal means hide this difference.
Both have mean 10; B is much more spread out.
Try it yourself.
Check your understanding
Find median of 8, 2, 6, 4.
Show answer & reasoning
5.
Order the data as . There are four observations, so the median is the average of the middle pair: .
Keep practising
6 worksheet questions · 12 marks
Read the questions online
- Find mean, median and range for 4, 6, 6, 8, 11.2 marks
- A long tail extends to high values. Name the skew.2 marks
- Groups share median 15 but ranges 4 and 20. State one comparison.2 marks
- Why use median for strongly right-skewed income?2 marks
- Five observations have mean 12. Find their total.2 marks
- Find the mode of 2, 2, 3, 5, 5, 6.2 marks
Read the worked solutions
Equivalent correct methods earn the corresponding marks. Include units where asked.
- Find mean, median and range for 4, 6, 6, 8, 11.2 marks
Total 35 divided by count 5; third value 6; .
Mean 7; median 6; range 7. - A long tail extends to high values. Name the skew.2 marks
The tail points towards larger values.
Positive/right skew. - Groups share median 15 but ranges 4 and 20. State one comparison.2 marks
The medians match; ranges differ.
The second group has wider overall spread. - Why use median for strongly right-skewed income?2 marks
The best summary also depends on the question.
It is less affected by a few very high incomes than the mean. - Five observations have mean 12. Find their total.2 marks
Mean multiplied by count.
60. - Find the mode of 2, 2, 3, 5, 5, 6.2 marks
Each occurs twice, the highest frequency.
Two modes: 2 and 5.
Curriculum reference & lesson scope
SACE Board of South Australia
Stage 1 General Mathematics: statistical investigation, univariate data, sampling, displays, summaries and interpretation.
Official curriculum source ↗